JC maths

Calculate Percentage Of A Given Number

(Given number/Total number) X 100.

nominal data

is categorical data which cannot be ordered.

e.g. makes of cars in a car park.

you cannot say a ford mondeo is greater than a renault clio. There is no order to the relationship.

here's a really good video explanation of data types:
https://www.youtube.com/watch?v\=--r9_R60Jws

ordinal data

is categorical data which can be ordered

e.g. exam grades, race results etc. We know that someone who gets an A has more marks than someone who gets a B, so there is an order to the data.

here's a really good video explanation of data types:
https://www.youtube.com/watch?v\=--r9_R60Jws

discrete data

only certain limited values are allowed. E.g. shoe sizes. you can't have a shoe size of 3.45961 for example. allowable sizes may be 1, 1.5, 2, 2.5, 3 ...

here's a really good video explanation of data types:
https://www.youtube.com/watch?v\=--r9_R60Jws

continuous data

constantly changing measurements e.g. height, weight , temperature. Where you can have very small infinitesmal changes in value. You are not limited to just a small set of possible values, like you are in shoe sizes for example(discrete numerical data).

here's a really good video explanation of data types:
https://www.youtube.com/watch?v\=--r9_R60Jws

central tendancy

mean, mode and median

all of these measurements give us an idea of the average of the data. Where the centre of the data lies

.mode

the element that appears most frequently in your data.

think of the word for fashion (mode). What does fashion mean - the cloths that people most often wear

.median

this is the data element in the middle (kind of sounds like middle)

sometime you will have 2 elements in the middle. In that case you have to get the average of the 2

.mean

Although the mean, the median and the mode are all types of averages. The mean is closest to what we normally associate with average.
i.e. add all the elements up and then divide by the count of the elements

.variability

- range, quartiles and interquartile
range

This gives us an indication of how spread out the data is.

For example take the situation where one team scores 2 goals in every match and another scores anything between 0 and 6 goals. Lets say the mean of both teams is 2. From this you might get the impression that the teams are similar. But if you check the ranges of each team, you will see that the second team has a much bigger range 6-0 \=6. As the first team scores 2 goals in every match, it has a range of 2-2 \= 0.

From this, we can say that the second team is much more inconsistent than the first team. So knowing the range helps us describe the data and compare different results

.outlier

extreme values that are not typical of the values in the data set. Far removed from the core set of data.

This is one of the main reasons why we have different types of averages. i.e. if data has outliers, it's better to use median or mode then to use mean as a way of determining average or central tendency as outliers will distort the mean result

.sample space

list of all possible outcomes

1. use systematic listing when a small number of outcomes is possible e.g tossing 2 coins { ht, hh, th, tt }.

2. use two way tables if there are many possible outcomes.

3. use tree diagrams for 2 or more events

e.g. toss coin followed by rolled die (dice)

trial is

the act of doing an experiment in probability.

e.g. throwing a die (dice) and recording which face it lands on

likelihood scale

impossible unlikely evens likely certain

relative frequency

is an estimate of the probability of an event based on the results of an experiment

fair or unbiased experiment

where all outcomes are equally likely to happen e.g. rolling die(dice).

number 2 is just as likely as the number 6 unless the die has been tampered with in which case the die is not fair

.principle of Counting

E.g. if one event has 2 possible outcomes (coin toss) and another has 6 possible outcomes(die roll) then the total number of possible outcomes is 2 x 6 \= 12

expected frequency

number of trials multiplied by relative frequency or probability

for example, if you do a trial of a hundred throws of a dice and you see that the number 6 appears 13 times, then the relative frequency would be 13/100 or 0.13.

Now lets say your are asked how many 6s' would you be expected to get if you throw 500 times, you would say 0.13 times 500. This would be the expected frequency

.theorem

is a rule or statement that you can prove by following a certain number of logical steps or by using a previous theorem or axiom that you already know

.proof

is a series of logical steps which we use to prove a theorem

axiom

is a statement that is accepted without any proof. We use axioms as building blocks to help logically prove geometry statements.

For example when proving that a triangle's angles add up to 180 degrees, we state at one stage in our argument that the angles in a line add up to 180 degrees. We don't have to prove this, it is accepted without proof

.corollary

is a statement that follows readily from a previous theorem

e.g. the corollary that an angle on a semi circle is always 90 degrees follows from the theorem that the angle at the centre of centre of a circle (180 degrees) is twice that angle at the circumferance (90) when both angles are standing on the same arc

.converse

the converse of a theorem is the reverse of a theorem

e.g. a theorem states that in an isosceles triangle the angles opposite the equal sides are equal in measure.
the converse of this would be that if two angles are equal in a triangle, then the triangle is isosceles

.what is the relationship between the slopes of perpendicular lines

e.g. if slope of one line is 3/4 then the slope of the line perpendicular to that line is -4/3 (rule flip and change sign).

products of their slopes are also \= -1.

i.e. 3/4*-4/3 \= -1 (always)

N or Natural numbers

{ 1,2,3,4,5...} - whole POSITIVE numbers \> 0

.W or Whole numbers

{ 0,1,2,3,4,5...} - Whole POSITIVE numbers including 0

.Z or Integers

{ ...-3, -2, -1, 0, 1, 2, 3...} - Whole positive AND negative numbers including 0

.Q

The set of Rational numbers that can be written as a fraction where the numerator and the denominators are both integers and the denominator is NOT 0

R

Real numbers - the set of all rational and irrational numbers

This basically includes all possible numbers. It can be fraction, decimal, whole number, negative, surd and any number you can think of. This is why we represent it as a thick line on the number line

R/Q: the set of
irrational
numbers

Numbers that cannot be written as fraction. e.g. pi or root 3, root 5 , root 7 etc

COMPLEMENT

the elements that are not in the stated set
Example: A' is read ''complement of A'' and means all the elements that are not in A

(all the elements outside of A)

distributive property

A(B+C) \= AB + AC

associative property

(AxB)xC \= A x (BxC)

for example if you have 2 times 4 times 5

you can evaluate this by multiplying the first 2 numbers (2 by 4)first to get 8 and then multiplying 8 by 5 to get your answer (40).

you can also evaluate it by multiplying the last 2 numbers first (4*5) and then multiplying your answer (20) by the first number 2 to get 40 also

.commutative property

8 + 3 \= 3 + 8


Adding and Multiplying numbers in a different order will not change the value of the sum or product


For example, addition is commutative (a+b\=b+a), but subtraction is not (a-b≠b-a)

.reciprocal

e.g. reciprocal of 3/4 is 4/3 (flip the fraction).

reciprocal of 4 is 1/4 (4 is 4/1, so flip it to get 1/4)

3⁻²

1/3²

scientific notation

A representation of real numbers as the product of a number between 1 and 10 and a power of 10, used primarily for very large or very small numbers. See Diagram for explanation of how to convert

.in scientific notation what does base number, coefficient and exponent stand for?

mark up

profit as % of cost price

margin

profit as % of selling price

gross tax

standard tax + higher tax

income tax paid

gross tax - tax credits

net tax

gross tax - tax credits (same as income tax paid and tax due)

1 kilogram \= ? grams

1000 grams

1 metre \= ? cms

100 cms

1 litre \= ? cm cubed

1000

set

a well-defined
collection of objects or elements

.2 sets are equal when

equality
is a relationship in which two equal
sets have the same elements

hypotenuse

The side of a right triangle opposite the right angle and the longest side of a right triangle

supplementary angles,

2 angles whose sum is 180 degrees

straight angle

An angle that measures 180 degrees

.transversal line,

a line that intersects two parallel lines

vertically-opposite angles

These are equal angles, opposite each other, formed when two straight lines intersect

.domain

All of the input or x values in a function

codomain

The set containing all the possible outputs of the function

range

All of the output or y values in a function

Can I use Pythagoras to find the slant height
(length), radius or height of a cone?

Yes as l h and r form a right angled triangle

Cardinal number of a set - n(S) or \#

The cardinal numbers is the number of elements in a set
if D\= {2,3,4,5,6,7,8,9,} then the n(D)\=8 or \# \= 8

.write down, state

means you don't have to show any workings. Though, you can if you so wish

Prove, calculate, find, show that, determine, prove

means you have to show workings... steps you took to get to the answer

.Solve

find the value of one or more variables in an equation.

Solve the equation\=get the roots of the equation

.Evaluate

substitute numbers in for variables in an expression and work out what the expression adds up to

Comment on

give your opinion on given information or answers

Construct

Draw very accurate diagram using pencil, ruler, set square, compass and protractor. Label carefully, leave constructions on your diagram, e.g. arcs of compass

.sketch

draw a rough diagram of a graph. Label if necessary. Doesn't have to be super accurate like constructions

.hence

use the answers from the previous section of the question to help you answer the current section

give your answer in the form

your answer has to look like this

e.g. you could be asked to work out the equation of a line and give your answer in the form y \= mx +c

so y -2x -3 \= 0 would not be in this form, you would have to change it to y \= 2x + 3 to match the required form

you need to be careful with this as you can lose marks for not giving our answer in the correct form.

Lets say you were told to give your answer in surd form and you got an answer root 3. If you were to go ahead and convert that root 3 (a surd) into a decimal, you would actually lose some marks

.express q in terms of z and w

you want to get q on its own equal to an expression with z and w in it.

eg 2q + z \= 3w
... 2q \= 3w -z
... q \= 3w-z/2

make q the subject of the equation

you want to get q on its own equal to an expression with z and w in it.

eg 2q + z \= 3w
... 2q \= 3w -z
... q \= 3w-z/2

Find the volume of the cylinder in terms of pi

you need to have pi appear in your end answer. For example, if you get a volume of pi (2²)(3)
you multiply the numbers and leave pi as it is. ie pi (12) \= 12pi

what should you check for first when asked to factorize an algebraic expression

Check if there is a common factor. If there is, then do Highest Common Factor method

what does a linear equation look like and how do you solve one

x + 4 \= 10 is an example. When the highest power of x is 1 then it is a linear equation.
Solve by putting letters to the left and numbers to the right until you get what x is.

x \= 10-4
x \= 6

x +y \= 4 is also a linear equation, but you will need an extra equation to solve it using simultaneous equation method

.what does a quadratic equation look like and how do you solve it?

x²+2x \= 4 is an example. The highest power of x is 2.

to solve it, you need to put all terms to the left of the \= sign and put equal to 0. Then factorize the left hand side, setting both factors \= 0 and solving for x.

note 3x² +x \= 0 is also a quadratic. You don't always have to have 3 terms

.what does an exponential function look like

eg f(n) \= 2ⁿ or 3(2ⁿ ) or 3ⁿ (junior cert course limited to 2ⁿ and 3ⁿ )

verify your answer

some times you are asked to solve an equation and then verify the answer you get. This means that you have to substitute the answer you get (usually an x value) into the original equation and see if it works out:

list of elements v number line

sometimes after solving an inequality ( eg -4 < x < 1)
you might be asked write the solution on the numberline or list the possible values of x ... for the latter you need to write your answer like this {-3,-2,-1,0}

when do you use a thick line on the number line

only when you're told that x ∊ R . If x is a real number, that means it can be any number on the number line, including decimal values. The thick line tells us that x can be any number along the line.

We use dots for x ∊ Z and x ∊ N as integers and natural numbers can only be whole numbers

.what does a quadratic sequence look like

0,5,8,9,8,5,0

notice if you graph one, you get either a u shape or an n shape

what does an exponential sequence look like

2,4,8,16,32

notice if you graph it, you get a curve that slopes only upwards.

in the example above you are multiplying by 2 for each term, so the gap between the numbers is getting increasingly wider

find nth term of a sequence

Let's say you have the following linear sequence: 2,5,8,11,14 and you wanted to figure out the 305th term in this sequence. You can do this using the nth term fomula ...Tn \= 3n-1. Tn stands for the value of the nth term. So T305\= 3(305) -1 \= 914

.what is the difference between range and codomain

In a function each input value has a corresponding output value. The set of these output values is called the Range.
The set of all possible output values is called the codomain. This may include some output values that are not mapped from an input value. e.g. {1,2,4,6,8,10} in the attached image which are part of the codomain but outside the range

.In a domain- codomain mapping if one of the domain elements has no arrow mapping to the codomain, is it a function?

no. each domain element must have a corresponding range element. I.e. each domain element must have an arrow pointing from it to the codomain

.f(x) \= x² - 2x find f(t-1)

always replace x with what is in the brackets (in this case t-1 is in the brackets)

f(t-1) \= (t-1)² - 2(t-1)
\= (t-1)(t-1) - 2(t-1)
\= t²-2t+1 - 2t +2
\= t² -4t +3

ray

is a part of a line that has one endpoint only. The other end goes on to infinity

.collinear points

all points are on the same line together. They are inline with each other

.what are measures of central tendancy

mean, median and mode

different ways to measure the average value of data

when to use the mode

if data is categorical. e.g. colour hair of students. You can't get the mean of hair colours as these are names which can't be added.

mode can also be used for number data.

use it also if you want to take out the effect of extreme outliers

.when to use the median.

only with numerical data

use it also if you want to take out the effect of extreme outliers

.when to use the mean

only with numerical data

not ideal if there are extreme outliers in the data. In that case mean doesn't give a true reflection of the average, as the outlier distorts often your results

.how to find sin cos and tan of 30 degrees and 60 degrees without using a calculator

draw equilateral triangle with sides \= 2cm

draw line down centre to c, so that it splits the base in half.

you can now find |ac| by using pythagoras theorem. Once you have the lengths of all 3 sides of the triangle on the left, then you can work out what sin cos and tan of 30 degrees and 60 degrees are without the calculator. In other words you can simply use the ratio of the sides i.e. opposite/hypotenuse, adjacent/hypotenuse, opposite/adjacent

.how to find sin cos and tan of 45 degrees without using a calculator

draw right angled isosceles with equal sides being 1 unit in length.

If you use pythagoras you will see that the length of the third side is root 2.

The 2 angles opposite the equal sides have to be equal to each other and so must be equal to 45 degrees. (90/2)

lower quartile value

the value that is one quarter way up the data set.

e.g. for dataset 1,2,2,3,4,5,5,6,7,7,9,11,12

there are 12 elements, so 1/4 of 12 is 3. This means that the lower quartile value is the average between 3rd and 4th value (2 and 3). i.e. 2.5

if we add one element onto the dataset as follows:
1,2,2,3,4,5,5,6,7,7,9,11,12,13

there are 13 elements, so LQ value is 1/4 of 13 \= 3.25
If you get a decimal answer like this, you should round up to 4, so the 4th value is the LQ value , in this case 3

.When should you use the Inter quartile range?

If there are outliers in the data, then range is not a good measurement of spread. For example, if you did a survey of the average scores of players on a football team, and all the team averaged between 1 and 2 goals, but one player for some reason is down as scoring an average of 20 goals. (maybe a typo)
So if you use the range to measure spread for this team, you might get the impression that this team is scoring a huge amount of goals, as the range is 20 - 1 \= 19.
Of course this isn't a true reflection of what the team is really doing.
To avoid this, we use the inter quartile range instead to measure spread. Any outliers that exist will be excluded from this calculation and so give a truer picture of what's going on

.similar triangles are

triangles where all 3 corresponding angles are equal.

remember that this does not mean that the sides are also equal.

also called equiangular triangles

congruent triangles

triangles in which all the corresponding sides and angles are equal.

the areas are also equal.

the only difference may be the direction and position of each triangle

.what does factorize mean

Get the factors of something.

e.g. factorize 12 \= 1(12) or 2(6) or 3(4) or 2(2)(3)

i.e. numbers that multiply to give your the thing you are trying to factorize.

you can also factorize algebraic expressions

e.g. factorize 2x +4 \= 2(x+2)

i.e. 2 expressions that multiply to give your the expression you are trying to factorize

.Axis of symmetry

a line that divides a shape into 2 perfectly symmetrical shapes

.center of symmetry

A translation is

when you move a point or a shape in a straight line. It moves every point in the same direction and the same distance